Binary Representation of Information 



29 



BINARY NUMBER REPRESENTATION 



If we recall the definition of our standard positional notation 

 whereby the meaning of a digit depends on its position relative 

 to other digits in the number representation, then we note that 

 any positive integer may be written 



(/„ . . . r/.//ir/o = r/o/i" + (JiB' + f/,/i' + . . . + daB" 

 where i:? > 1 is the base of the number representation and where 

 ^d,< B. 



For example, if Z? = 10 the integer "fifty-seven" may be written 

 57 = 7 . 10' + 5 . kV 

 - 7 + 50. 

 \i B = 1 then "fifty-seven" becomes 



Llll0011,,vo = [2'^ + 2' + 2' + 2i,,.„ 

 = [1 + 8 + 1(> + 32],,„ 



where the subscript "two" indicates that binary notation is used 

 on the left of the equal sign and the subscript "ten" indicates 

 that decimal notation is used on the right. 



Similarly, any positive fraction (less than one) may be written 



OV/_if/_or/^3 . . . d-,n = (UB-' + r/_o^-' + d.^B'^ + . . . + r/_„,fi""', 

 where m does not have to be finite and ^ «'_, < B. For example, 

 if /? = 2 then fi\e-sixteenths becomes 



[0.0101]uvo = 



2"' + 2- 

 1 + i,' . 



.4 IbJten 



Since positive numbers may be decomposed into an integer part 

 and a fraction part we may consider, as a more general example, 

 the binary representation of thirty-seven and nine sixty-fouiths. 



[I00101.0010011t„,, = [2" + 2' + 2' 



,„ + [2-^ + 2^^J 



[1 + 4 + 82],,„ + 



1+^ 

 L8 ()4J 



[37]ten + 



!) 

 L()4jt 



It is not our intention to go into great detail at this point and 

 discuss the procedures for converting from one number representa- 

 tion to another. We have merely tried to review, by means of 



